nLab principal Sp(2)-bundle

Idea

Principal Sp(2)-bundles (or principal Spin(5)-bundles) are special principal bundles with the second symplectic group Sp(2) (exceptionally isomorphic to the fifth spin group Spin(5)) as structure group (gauge group).

Examples

• One has $S^{4n+3}\cong Sp(n+1)/Sp(n)$, hence there is a principal Sp(2)-bundle $Sp(3)\twoheadrightarrow S^11$. Such principal bundles are classified by:
  $$\pi_11B Sp(2) \cong\pi_10 Sp(2) \cong\mathbb{Z}_120.$$

  (Mimura & Toda 63)

Related concepts

Particular principal bundles:

• double cover (principal O(1)-bundle)
• principal O(2)-bundle
• principal U(1)-bundle (principal SO(2)-bundle)
• principal U(2)-bundle (principal $Spin^\mathrm{c}(3)$-bundle, principal $Spin^\mathrm{h}(2)$-bundle)
• principal U(3)-bundle
• principal SO(3)-bundle
• principal SO(4)-bundle ($Spin^\mathrm{h}(3)$-bundle)
• principal SO(5)-bundle
• principal SO(6)-bundle
• principal SO(7)-bundle
• principal SO(8)-bundle
• principal SU(2)-bundle (principal Sp(1)-bundle, principal Spin(3)-bundle)
• principal SU(3)-bundle
• principal SU(4)-bundle (principal Spin(6)-bundle)
• principal Sp(2)-bundle (principal Spin(5)-bundle)

References

• Mamoru Mimura and Hiroshi Toda, Homotopy Groups of SU(3), SU(4) and Sp(2) (1963), Journal of Mathematics of Kyoto University. 3 (2), p. 217–250, doi:10.1215/kjm/1250524818